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A347580 Triangle read by rows: T(n,k) is the number of chains of length k in the poset of all arithmetic progressions contained in {1,...,n} of length in the range [1..n-1], ordered by inclusion.

%I A347580 #37 Sep 13 2021 11:38:12
%S A347580 1,1,2,1,6,6,1,12,24,12,1,21,68,72,24,1,32,144,244,180,48,1,47,283,
%T A347580 666,764,432,96,1,64,486,1510,2436,2164,1008,192,1,85,799,3117,6534,
%U A347580 8028,5816,2304,384,1,109,1232,5860,15368,24524,24516,15040,5184,768
%N A347580 Triangle read by rows: T(n,k) is the number of chains of length k in the poset of all arithmetic progressions contained in {1,...,n} of length in the range [1..n-1], ordered by inclusion.
%C A347580 Let L_n be the lattice of all arithmetic progressions contained in {1,...,n}, including the empty progression and the whole interval. T(n,k) is the number of chains of length k+2 in L_n that contain both the maximal and minimal element.
%H A347580 M. K. Goh, J. Hamdan, and J. Saks, <a href="https://arxiv.org/abs/2106.05949">The lattice of arithmetic progressions</a>, arXiv:2106.05949 [math.CO], 2021. See Table 2 p. 7.
%F A347580 Let f(n,k) = n, if k=1; A338993(n,k)/2, if 2<=k<=n. Then T(n,k) = 1, if k=1; Sum_{i=1..n-1} f(n,k)*T(i,k-1), if 2<=k<=n; 0, if k>n.
%F A347580 Sum_{k=1..n} (-1)^k*T(n,k) = A008683(n-1), for n>=2.
%e A347580 Triangle begins:
%e A347580   n/k 1   2    3     4     5      6      7      8      9    10    11   12
%e A347580    1  1
%e A347580    2  1   2
%e A347580    3  1   6    6
%e A347580    4  1  12   24    12
%e A347580    5  1  21   68    72    24
%e A347580    6  1  32  144   244   180     48
%e A347580    7  1  47  283   666   764    432     96
%e A347580    8  1  64  486  1510  2436   2164   1008    192
%e A347580    9  1  85  799  3117  6534   8028   5816   2304    384
%e A347580   10  1 109 1232  5860 15368  24524  24516  15040   5184   768
%e A347580   11  1 137 1838 10418 33049  65402  84284  70992  37760 11520  1536
%e A347580   12  1 167 2611 17420 65706 157010 250332 270996 197280 92608 25344 3072
%t A347580 t[n_, k_] := If[k == 1, n, Sum[2(n-(k-1) r), {r, 1, Quotient[n-1, k-1]}]];
%t A347580 f[n_, k_] := If[k == 1, n, t[n, k]/2];
%t A347580 T[n_, k_] := T[n, k] = If[k == 1, 1, Sum[f[n, i] T[i, k-1], {i, 1, n-1}]];
%t A347580 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 13 2021, from PARI code *)
%o A347580 (PARI) t(n, k) = if (k==1, n, sum(r=1, (n-1)\(k-1), 2*(n-(k-1)*r))); \\ A338993
%o A347580 f(n, k) = if (k==1, n, t(n,k)/2);
%o A347580 T(n, k) = if (k==1, 1, sum(i=1, n-1, f(n, i)*T(i, k-1))); \\ _Michel Marcus_, Sep 11 2021
%Y A347580 Cf. A008683, A338993.
%K A347580 nonn,tabl
%O A347580 1,3
%A A347580 _Marcel K. Goh_, Sep 07 2021